When an electric dipole is held at an angle in a uniform electric field, the net force F and torque `tau` on the dipole are

Options

A. `F = 0, tau = 0`
B. `F != 0, tau != 0`
C. `F = 0, tau != 0`
D. `F != 0, tau = 0`

Explanation

To solve the problem of finding the net force \( F \) and torque \( \tau \) on an electric dipole held at an angle \( \theta \) in a uniform electric field \( E \), we can follow these steps:

### Step 1: Understand the Electric Dipole
An electric dipole consists of two equal and opposite charges, \( +Q \) and \( -Q \), separated by a distance \( L \). The dipole moment \( \mathbf{P} \) is defined as:
\[ \mathbf{P} = Q \cdot \mathbf{L} \]
where \( \mathbf{L} \) is a vector pointing from the negative charge to the positive charge.

### Step 2: Analyze the Forces on the Dipole
When the dipole is placed in a uniform electric field \( \mathbf{E} \):
- The positive charge \( +Q \) experiences a force \( \mathbf{F_1} = +Q \mathbf{E} \) in the direction of the electric field.
- The negative charge \( -Q \) experiences a force \( \mathbf{F_2} = -Q \mathbf{E} \) in the opposite direction.

### Step 3: Calculate the Net Force
The net force \( \mathbf{F} \) on the dipole is the vector sum of the forces acting on the charges:
\[ \mathbf{F} = \mathbf{F_1} + \mathbf{F_2} = Q \mathbf{E} - Q \mathbf{E} = 0 \]
Thus, the net force on the dipole is zero:
\[ \mathbf{F} = 0 \]

### Step 4: Calculate the Torque
The torque \( \tau \) on the dipole due to the forces can be calculated using the formula:
\[ \tau = \mathbf{r} \times \mathbf{F} \]
Where \( \mathbf{r} \) is the position vector from the pivot point (midpoint of the dipole) to the point of application of the force.

For the positive charge \( +Q \), the distance from the center to the charge is \( \frac{L}{2} \) and for the negative charge \( -Q \), it is also \( \frac{L}{2} \). The torque due to each force can be calculated as follows:
- For \( +Q \):
\[ \tau_1 = \left(\frac{L}{2}\right) \cdot Q E \cdot \sin(\theta) \quad \text{(clockwise)} \]
- For \( -Q \):
\[ \tau_2 = \left(\frac{L}{2}\right) \cdot Q E \cdot \sin(\theta) \quad \text{(counterclockwise)} \]

Since both torques are in the same direction (clockwise), the net torque \( \tau \) is:
\[ \tau = \tau_1 + \tau_2 = Q E L \sin(\theta) \]

### Final Results
Thus, the results are:
- The net force \( F \) on the dipole is:
\[ \mathbf{F} = 0 \]
- The net torque \( \tau \) on the dipole is:
\[ \tau = Q E L \sin(\theta) \]